In previous work on quota formulas at the Fund, the choice of weights lacked objectivity as proposed alternative formulas aimed at achieving specific objectives such as a predetermined country ranking (see Mirakhor, 2006) or replication of prevailing quota distributions (see IMF, 2000). In addition, weights assigned to variables in proposed Fund quota formulas outside the IMF were generally the result of authors’ judgment and not that of a statistical method.
We argue that a significant part of judgment is avoided when the PCA is used to generate the weights of the variables entering a new formula. Indeed, weights are determined by the analysis itself, given the data set provided, irrespective of any other consideration. No
reference is made to existing quota formulas or prevailing quota distributions in that process.
Hence, the PCA brings some objectivity about the relative weights of the variables, which could serve as starting point to advance discussions on a definitive set of weights.
Quota formula
As indicated earlier, the first principal component is always the weighted linear composite of the original variables with weights chosen so that the composite accounts for the maximum variation in the original data. Using the weights of this component, a quota formula is derived as follows:
Quota formula: α1*x1 + α2*x2 +…+ αp*xp (1)
where αi= wi / (w1 + w2 +…+ wp) with w1, w2,.., wp the weights determined by the PCA and x = (x1, x2,…, xp) is the set of the original variables expressed in terms of countries’ shares in global totals.
With variables expressed in terms of countries’ shares in global totals (as in the QFRG) and the sum of the weights equal to one, the output of the formula is not the amount of quotas but the quota shares, which sum to one. The quota formula is therefore linear in shares. In
addition, it keeps invariant the ranking of the IES.
Properties of the quota formula
As expressed in equation (1), the quota formula has all the characteristics the Executive Board of the Fund requested regarding the specification of a new quota formula, that is (see IMF 2001):
• simple and transparent as it is parsimonious in the number of variables; and
• homogeneous in the sense that a uniform change in all variables leaves calculated quotas shares unchanged.
Assessment criteria for the quota formula
The acceptance of a quota formula faces two types of constraints: political and technical.
Some political constraints are explicitly expressed in the Resolution. As regards the weights of the variables, the Resolution provides that in devising a new quota formula “consideration should be given to placing higher weight on members’ gross domestic product, together with ensuring that other variables, in particular the openness of member countries, also play an important role.” Therefore any quota formula should respond positively to this provision i.e.
ensure that the GDP variable has the highest weight in the new quota formula. Moreover, there exist other constraints, tough not explicitly formalized in an official Fund document but repeatedly mentioned that need to be taken into account when developing a new formula.
Among these constraints are:
• the quota share of the major shareholder should not be lower than 15%;
• emerging market economies (EMEs) quota shares should increase; and
• a new quota formula should lead to a rebalancing of quota shares from advanced countries to developing and transition countries;
The technical constraints relate more to the consequences of a new formula on the size of the Fund. Indeed, as it is generally agreed (since the conclusion of the 12th General Review of Quotas) that the size of the Fund is adequate, a new quota formula should not require a significant increase in the total of quotas for the resulting quota distribution to be approved by the membership as the definitive applicable distribution. Hence, one constraint will be the minimum rate of increase in total quotas, defined as the maximum of all rates that would at least maintain individual countries’ level of quotas –assuming that no country would accept a decline in its level of quotas. A moderate rate will be considered acceptable.
Moreover, the acceptance of a new quota formula by the membership could prove difficult if it entails significant changes in the quota structure and in rankings. The average change in ranking, measured as the root square of the squared differences between the IES ranking and that of the actual quota structure, is proposed to assess the degree of change in the ranking.
III. APPLICATION
A. Variables
As noted earlier, there have been many discussions on the variables to be included in a new quota formula. Some Directors would like four variables to be considered while others Directors think that fewer variables be included. In particular, while all Directors agreed that GDP must be included in the new quota formula, there was less an agreement regarding whether it should be converted in market exchange rate or in purchasing power parity (PPP).
Likewise, the variability measure as it is proposed has been criticized and alternative measures of the variability have been proposed (see for example dos Reis, 2005). The proposed openness variable has been seen by some Directors as a duplicate of the GDP, and adjustments were suggested to include financial openness to better reflect integration in the global economy. Its inclusion in the formula has even been questioned as it favors mostly countries with high values in these variables. There has been a debate on whether keeping Reserves in the formula, as some Directors are of the view this variable is no more an indication of a country’s ability to contribute to the resources of the Fund.
To illustrate the principal components approach, different sets of data with the four variables abovementioned but measured differently will be explored. In particular, two new measures of GDP and variability never proposed before are introduced:
• a hybrid GDP measure (HGDP) which is defined for each country as the highest of the GDP measured at market exchange rate and the GDP converted in PPP (PPPGDP).
• a hybrid variability variable (HVAR) defined as the highest of the variability of currents receipts (VREC) and that of the variability of current receipts and net capital inflows (VAR)
Since some countries expressed a preference to the PPPGDP over the traditional GDP, and other favour the traditional GDP, it seems logical that the hybrid GDP as proposed leaves each country the opportunity to choose the GDP measure that could increase their quota shares. The hybrid variability is justified on the grounds that for some countries, mainly commodity-exporting countries, the potential need for Fund resources that variable is deemed to reflect is more likely from current account shocks than from capital account shocks. Hence
letting the countries choosing the type of measure that reflects the most their potential need for Fund resources seems a best option.
The data sets that will be used are:
• Data Set I (DS I) : GDP, Reserves (RES), Openness (OPEN) and VAR
• Data Set II (DS II) : GDP, RES, OPEN and VREC
• Data Set III (DS III) : GDP, RES, OPEN and HVAR
• Data Set IV (DS IV) : HGDP, RES, OPEN and VAR
• Data Set V (DS V) : HGDP, RES, OPEN and VREC
• Data Set VI (DS VI) : HGDP, RES, OPEN and HVAR
Data are provided by the Fund. These data are those available at end-2004 and concern the whole membership of the Fund (184 countries) as of end-2004 (Statistical Appendix I). All IMF statistical methodologies and terminologies are implicitly adopted in this paper. A distribution of all variables with respect to regional country groupings is presented in Table 1 below. Countries’ classifications are detailed in Appendix V.
B. The Indicator of Economic Size Interpretation of the output of the PCAs
Table 2 shows that the proportion of total variance accounted for by the first principal component for each of the data sets is high, between 75 and 77 percent5. This means that the first principal component objectively explains 75 to 77 percent of the differences among countries and that 23 to 25 percent of the information contained in the original variables is lost by aggregating them in one composite variable. Nevertheless, the first component clearly indicates an overall size dimension among the countries; which in our case, given the nature of our variables, could be interpreted as the country’s relative economic size or importance in the global economy.6 Therefore, this principal component will serve as the indicator of economic size and the proportion of total variance it explains indicates a high level of objectivity of the indicator.
5 PCAs have been performed with the SPAD software. More information on this software is available at www.spadsoft.com. PCAs can also be performed in other softwares such as Eviews, SPSS, Givewin.
6 The second component in such circumstances highlights contrasts between the countries. As Reserves is highly correlated with that component in all six cases, it therefore displays contrast between the most important countries, with respect to the level of reserves. See Appendix II for a more detailed analysis.
Table 1. Distribution of variables in world totals by country groupings 2004 2002-04 2004 2002-04 2002-2004 2004 2000-04 2000-04 2000-04 1992-2004 1992-2004 1992-2004
Advanced economies 61.59 60.51 67.08 76.27 76.90 50.49 51.53 52.48 43.35 70.89 69.13 70.01 61.34 58.14 59.94
Major Advanced (G7) 46.03 45.22 47.32 63.64 64.88 42.36 43.22 43.85 34.36 50.32 47.66 49.00 43.12 36.86 40.53
USA 17.38 17.08 16.80 28.72 30.35 20.47 20.69 20.26 2.71 18.14 13.42 15.80 20.37 11.83 17.37
Other advanced 15.56 15.29 19.76 12.63 12.02 8.13 8.31 8.64 8.99 20.57 21.47 21.02 18.22 21.27 19.40
Developing countries 30.87 32.08 27.61 20.00 19.72 43.29 42.38 41.55 48.80 24.01 25.58 24.79 31.98 34.93 33.46
Africa 5.49 5.40 2.43 1.68 1.56 3.37 3.37 3.29 3.27 1.96 2.10 2.03 4.11 4.42 4.39
Asia 10.29 11.50 15.27 10.38 10.28 28.63 27.74 27.17 33.19 13.37 14.31 13.84 13.52 15.14 13.25
Middle East 7.63 7.60 4.73 3.03 2.90 3.77 3.71 3.70 5.84 3.53 4.12 3.82 6.30 9.15 8.52
Western Hemisphere 7.46 7.59 5.18 4.91 4.98 7.52 7.55 7.39 6.50 5.15 5.06 5.11 8.06 6.22 7.30
Transition economies 7.54 7.41 5.31 3.72 3.38 6.21 6.09 5.97 7.86 5.11 5.29 5.20 6.68 6.93 6.60
Source: IMF
GDP PPPGDP
Table 2. Principal Component Analysis - Output
1st Principal Component Component Component Component Component Component Component Component Component Component Component Component
Eigenvalues (Variance) 13.01 3.59 12.39 3.45 12.45 3.53 10.98 3.08 10.61 2.86 10.53 2.95
Proportion of variance explained 0.76 0.21 0.76 0.21 0.76 0.22 0.75 0.21 0.76 0.20 0.75 0.21
Cumulative proportion 0.76 0.98 0.76 0.97 0.76 0.98 0.75 0.96 0.76 0.96 0.75 0.96
Variables highly correlated GDP RES GDP RES GDP RES HGDP RES HGDP RES HGDP RES
with the principal component OPEN OPEN OPEN OPEN OPEN OPEN
VAR VREC HVAR VAR VREC HVAR
Source: Author's calculations
Data Set IV Data Set V Data Set VI
Data Set I Data Set II Data Set III
Figure 2. Swarm of countries in the plane formed by the first two principal components
Source: Author
Figure 2 shows the swarm of countries in a plane formed by the first two principal components using the Data Set I. The two components account for 98 percent of the
differences among countries. The swarm indicates countries’ relative positions and those of the top 10 countries are indicated. Scores of the indicator are obtained by an orthogonal projection on the first principal component (which is the dotted horizontal line in the figure).
Ranking
The scores of the composite indicator are used to establish a ranking of the countries (Statistical Appendix II). Table 3 shows that the six rankings are broadly similar for the top 20 important countries.7 A striking finding is that the top 6 countries are the same in all data sets, with a slight but significant change compared to the actual ranking. While the two most important countries remain the same (US and Japan), China stands firm at the third rank in all data sets, confirming its strong dynamism in the international scene. In all sets, France exits the top 5 to stand at the 6th place, behind United Kingdom (5th) and Germany (4th).
Another finding is that the top 9 rankings are the same for data sets including GDP.
Sixteen countries remain in the top 20 in the six variants of our indicator. One country (Singapore) which has experienced a strong economic growth over the past decades is entering the top 20 in the six sets. Ireland enters the top 20 in four sets. Korea ranks at least 8th in any set from the 19th place in the actual ranking. Mexico experienced the second greater jump within the top 20 with at least 4 places gained and China is the third country to get the highest jump with a gain of 3 places.
The impact of the hybrid GDP is obvious with respect to the developing countries in the top 20. With the exception of Venezuela and Saudi Arabia, all developing countries have a better ranking in the top 20. In particular, India and Brazil gains at least 5 and 3 places respectively, in data sets with HGDP.
There is also a similarity of results when it comes to the most important gains in economic importance. The top 20 jumps show that actually four countries, namely Botswana,
Equatorial Guinea, Estonia and Turkmenistan, are among the five countries with the highest gains in all sets. In general, the reasons behind these top 20 gains are diverse ranging from sound macroeconomic policy implementation to discovery and exploitation of natural resources.
On the other hand, many factors including conflicts over the past decades may explain the most important declines in ranking. One could notice that most of the countries dropping in the ranking are from Africa.
7 If we consider that actual quotas reflect the position in the global economy.
Overall, the six rankings give a very good picture of countries relative importance in the world economy, especially in the top twenty spots. In addition, those rankings are quite similar, which allows us to conclude that the use of the PCA to establish countries’ relative economic weights is worth doing and hence, the IES, in its various specifications, is
objective.
C. Quota Formulas Weights of the variables
A look at variables weights in Table 5 shows that the GDP variable is assigned a higher weight by the PCA in all sets, which is due to the fact that this variable in all sets has the highest variance. 8 However, this weight varies only between 0.29 when the hybrid form of the GDP is used and 0.35 when GDP is expressed at market exchange rate.
Whatever the set, the weight of the openness variable is non negligible and is interestingly fixed at around 0.22-0.23. Similarly, the 3 Variability measures have the same weights in the two formulas they each enter in. For example, the current receipts and net capital inflows have a constant weight of 0.21 with either form of GDP. Consequently, Reserves appears as being given the residual weight, which varies from 0.20-0.23 when GDP at market exchange rate is used to 0.26-0.27 with HGDP. Overall, GDP has the highest weight in all formulas.
Openness has an important weight too, though this weight does not differ significantly from those of Variability and Reserves.
Quota distributions
The diverse sets of weights are used to calculate individual quota shares. Countries’ quota shares vary across formulas. Nevertheless, regarding the top 20 countries, the USA has lower than actual quota shares in data sets with HGDP and those shares are lower than 15 percent.
Among G7 countries, Japan is the only country always experiencing an increase in its quota share while those of France, UK and Italy always decline (see Statistical Appendix III). In the three data sets including GDP, Japan ends up with a doubling of its actual quota share. At least four out of the five countries that are recording the highest increases in their quota shares are from Asia, confirming assertions that this region is a rising economic power.
8 From an economic standpoint, GDP should probably have the highest weight, independently of whether the cross-country variability of GDP is higher or lower than that of the other variables in the data sets. However, the PCA would give the largest weight to the variable with the greatest cross-country variation –irrespective of other considerations–, which happens to be GDP in all our data sets.
Actual
1 United States United States 0 United States 0 United States 0 United States 0 United States 0 United States 0
2 Japan Japan 0 Japan 0 Japan 0 Japan 0 Japan 0 Japan 0
3 Germany China 3 China 3 China 3 China 3 China 3 China 3
4 France Germany -1 Germany -1 Germany -1 Germany -1 Germany -1 Germany -1
5 United Kingdom United Kingdom 0 United Kingdom 0 United Kingdom 0 United Kingdom 0 United Kingdom 0 United Kingdom 0
6 China France -2 France -2 France -2 France -2 France -2 France -2
7 Italy Italy 0 Italy 0 Italy 0 India 6 Korea 12 Korea 12
8 Saudi Arabia Korea 11 Korea 11 Korea 11 Korea 11 India 5 India 5
9 Canada Canada 0 Canada 0 Canada 0 Italy -2 Italy -2 Italy -2
10 Russia Mexico 6 Netherlands 1 Netherlands 1 Russia 0 Russia 0 Russia 0
11 Netherlands Spain 6 Mexico 5 Mexico 5 Canada -2 Canada -2 Canada -2
12 Belgium Russia -2 Russia -2 Spain 5 Mexico 4 Mexico 4 Mexico 4
13 India India 0 Spain 4 Russia -3 Brazil 5 Netherlands -2 Brazil 5
14 Switzerland Netherlands -3 India -1 India -1 Spain 3 Singapore 34 Netherlands -3
15 Australia Singapore 33 Singapore 33 Singapore 33 Singapore 33 Brazil 3 Singapore 33
16 Mexico Brazil 2 Switzerland -2 Brazil 2 Netherlands -5 Spain 1 Spain 1
17 Spain Switzerland -3 Belgium -5 Switzerland -3 Switzerland -3 Switzerland -3 Switzerland -3
18 Brazil Ireland 32 Australia -3 Belgium -6 Ireland 32 Belgium -6 Belgium -6
19 Korea Australia -4 Brazil -1 Australia -4 Indonesia 4 Malaysia 11 Malaysia 11
20 Venezuela Belgium -8 Malaysia 10 Ireland 30 Malaysia 10 Indonesia 3 Indonesia 3
Source: Author's calculations
Data Set V Data Set VI
Data Set I Data Set II Data Set III Data Set IV
Table 4. Indicator of Economic Size – The top 20 jumps in country rankings
Ranking Country Gain Country Gain Country Gain Country Gain Country Gain Country Gain
1 Botswana 63 Botswana 67 Botswana 63 Botswana 63 Botswana 68 Botswana 64
2 Estonia 49 Estonia 54 Estonia 50 Turkmenistan 46 Turkmenistan 53 Turkmenistan 48
3 Turkmenistan 45 Turkmenistan 51 Turkmenistan 49 Estonia 43 Estonia 49 Estonia 47
4 Equatorial Guinea 43 Albania 42 Equatorial Guinea 42 Equatorial Guinea 42 Equatorial Guinea 41 Equatorial Guinea 41
5 Albania 38 Equatorial Guinea 42 Albania 41 Ethiopia 65 Albania 65 Albania 65
6 Luxembourg 36 Luxembourg 36 Luxembourg 35 Cambodia 35 Luxembourg 36 Cambodia 35
7 Bahrain 35 Bahrain 36 Bahrain 34 Albania 35 Bahrain 36 Luxembourg 34
8 Singapore 33 Cambodia 35 Singapore 33 Luxembourg 34 Ethiopia 36 Bahrain 34
9 Ethiopia 33 Singapore 33 Lebanon 32 Singapore 33 Cambodia 36 Singapore 33
10 Ireland 32 Lebanon 33 Ethiopia 31 Bahrain 33 Singapore 34 Ethiopia 33
11 Lebanon 32 Ethiopia 33 Cambodia 31 Ireland 32 Nepal 34 Lebanon 31
12 Cambodia 31 Lithuania 31 Ireland 30 Lebanon 32 Lebanon 32 Nepal 30
13 Lithuania 27 Ireland 29 Lithuania 29 Nepal 29 Lithuania 29 Lithuania 27
14 Cyprus 24 Malta 29 Oman 26 Lithuania 25 Ireland 26 Ireland 26
15 Nepal 24 Oman 26 Cyprus 24 Bhutan 23 Oman 24 Oman 25
16 Macedonia, FYR 24 Macedonia, FYR 24 Malta 24 Turkey 21 Bhutan 24 Bhutan 24
17 Turkey 21 Latvia 23 Swaziland 24 Oman 21 Malta 22 Slovenia 20
18 Malta 21 Nepal 23 Macedonia, FYR 23 Jordan 20 Paraguay 22 Cyprus 20
19 Bhutan 21 Swaziland 23 Nepal 22 Cyprus 19 Burkina Faso 21 Malta 20
20 Oman 20 San Marino 23 San Marino 22 Macedonia, FYR 19 Swaziland 21 Burkina Faso 20
Source: Author's calculations
Data Set III Data Set IV Data Set IV Data Set IV
Data Set I Data Set II
No African country appears in the top quota increases but many of them are experiencing the most significant declines in quota shares. Another interesting finding is that many of those countries that see their quotas share shrinking are oil-producing countries. While the appearance of Kuwait or even Iraq could be understandable given the turbulences in that region, this finding may seem counterintuitive as one would expect them among the most dynamic countries given their significant natural resources.
Country groupings and Executive Board constituencies
The new quota structure in terms of country groups and Executive Board constituencies are shown in Table 5. The quota structures show an increase in the share of advanced economies when GDP at market exchange rate is used. Conversely, when the Hybrid GDP is included in the formula, advanced countries are losing quota shares in favor of developing countries –and hence, there is quota rebalancing. In all quota formulas, the transition countries as a group have a lower quota shares but this decrease is lower with HGDP.
The increase in the share of advanced economies is essentially owing to those of Japan, Ireland and Spain. In the developing and transition countries groups, many African, Middle East and Transition countries would lose a significant size of their quota shares. Conversely, Asian developing countries would benefit from the sharp increase in China’s share and those of Korea and Singapore. When the Hybrid GDP is included in the formula, the group of developing Asian countries at the IMF tops the United States.
Implications for the size of the Fund
Should any PCA-derived quota structure be retained as such, this would imply a significant increase in total quotas. As shown in Table 5, the size of the quotas would have to be at least 27 times higher than what it is presently. It would be the case because many countries with already a small quotas shares would have them further reduced, thus adding to the increase of
Should any PCA-derived quota structure be retained as such, this would imply a significant increase in total quotas. As shown in Table 5, the size of the quotas would have to be at least 27 times higher than what it is presently. It would be the case because many countries with already a small quotas shares would have them further reduced, thus adding to the increase of